
Using the Dirichlet-to-Neumann operator \(\Lambda \)  on differential forms, Belishev and Sharafutdinov in [1]} describe an ancillary boundary operator called the Hilbert transform \(T\) . The Hilbert transform is a classical operator in complex analysis and it also appears in Clifford analysis as an operator on the \(L^2\) -completion of the space \({M}{}\) . Two good sources include Brackx and De Schepper's paper [2]} (which specifically concentrates on compact regions of \(^n\) ) and Calderbank's thesis [3]}.
Given a function on the boundary \(\phi \in {M}{}\) , the Hilbert transform in Clifford analysis allows one to uniquely recover the complete boundary data of a monogenic field \(f\)  with \(\phi \)  as a component of \(f\vert _{\partial M}\) . In essence, the Hilbert transform yields boundary values of functions conjugate in the generalized Cauchy–Riemann equations given by \(f=0\) . In fact, if \(\phi _k\)  is a \(k\) -vector, the Hilbert transform of \(\phi _k\)  contains a \(k-2\) -, a \(k\) -, and \(k+2\) -vector component. This is part of why we suggest to look beyond pairwise conjugate forms.
A reasonable question to ask is: are these two Hilbert transform operators related in some way? Moreover, Sharafutdinov and Shonkwiler extend Belishev and Sharafutdinov's Dirichlet-to-Neumann operator to the complete Dirichlet-to-Neumann operator [4]}. If it is not possible to relate Belishev and Sharafutdinov's Hilbert transform using the DN operator on forms to the Hilbert transform in Clifford analysis, is there a related operator defined in terms of the complete DN operator that relates to Clifford analysis?

This point is essentially given as an open question by Belishev and Vakulenko [5]}. Specifically, those two ask whether it is true that the algebras \(\vee ^+(M)\)  and \(\vee \mathrm {tr}^+(M)\)  are isometrically isomorphic. At the moment, we have a partial answer: by the Cauchy integral formula, a monogenic field \(f\in (M)\)  is determined by its boundary values therefore the map \(\colon (M) \rightarrow (M)\)  is an isomorphism of vector spaces and by the weak maximum principle for elliptic operators, it is also an isometry. This of course applies to the spinor subspace \(^+(M)\) . However, it is not clear that the algebras \(\vee ^+(M)\)  and \(\vee \mathrm {tr} ^+(M)\)  are isomorphic. In Clifford analysis, we have the Hardy space \((M)\)  as the \(L^2\) -completion of \((M)\)  which is studied in both the previously referenced papers [2]}, [3]}. Perhaps there is more knowledge about \((M)\)  that could assist in finding a proof of fact (ii).

There is likely geometric content inside the spinor spectrum and this could lead to determining, at the very least, the conformal class of the metric. First, it is widely known that the Hodge–Dirac operator is conformally invariant [8]}. Hence, it may be possible to construct a metric \(g\)  up to conformal equivalence from the spinor spectrum given that the spinor spectrum is already homeomorphic to \(M\) . This should not be shocking; Belishev in [9]} was able to do this for surfaces \(S\)  with single boundary component, as he proved that the topologized spectrum is conformally equivalent to \(S\)  (and in fact was determined by the classical DN operator). It could be that an procedure analogous to Belishev's technique for finding a conformal metric in [9]} can be performed with the spinor spectrum.
It could be that we can do better than extracting just the conformal class for dimension \(n\ge 3\) . As a reminder, the 2-dimensional problem cannot determine more than the conformal class of \(g\)  since \(\Delta \)  is conformally invariant in dimension 2. But, if we consider subsurfaces \(S\)  inside of \(M\) , we can collect conformal copies of the metric restricted to the surface, vary the over a collection of surfaces passing through a point, and perhaps the combined data from all surfaces passing through each point could produce a metric in the isometry class of \(M\) .

